Yesterday I learned that geometric relations between events can be characterized generally (and up to a common non-zero factor) in terms of their pairwise "Lorentzian distance, $d_{\ell}$", which satisfies for any three events, $\mathsf A$, $\mathsf B$, $\mathsf Q$
if $\mathsf A$ and $\mathsf B$ are space-like or light-like related to each other then $d_{\ell}[~\mathsf A, \mathsf B~] = d_{\ell}[~\mathsf A, \mathsf B~] = 0$; and otherwise:
if $\mathsf A$ and $\mathsf B$ are time-like related to each other then $d_{\ell}[~\mathsf A, \mathsf B~]$ may be zero, finite, or infinite;
if $d_{\ell}[~\mathsf A, \mathsf B~]$ is finite then $d_{\ell}[~\mathsf B, \mathsf A~] = 0$; and
if $d_{\ell}[~\mathsf A, \mathsf B~]$, $d_{\ell}[~\mathsf A, \mathsf Q~]$ and $d_{\ell}[~\mathsf B, \mathsf Q~]$ are all finite then the inverse triangle relation holds:
$$\left( \frac{d_{\ell}[~\mathsf A, \mathsf B~]}{d_{\ell}[~\mathsf A, \mathsf Q~]} \lt 1 \right) \text{ and } \left( \frac{d_{\ell}[~\mathsf B, \mathsf Q~]}{d_{\ell}[~\mathsf A, \mathsf Q~]} \lt 1 \right) \implies \left( \frac{d_{\ell}[~\mathsf A, \mathsf B~]}{d_{\ell}[~\mathsf A, \mathsf Q~]} + \frac{d_{\ell}[~\mathsf B, \mathsf Q~]}{d_{\ell}[~\mathsf A, \mathsf Q~]} \le 1 \right).$$
Questions:
Given a spacetime $\mathscr S$ as a set of events together with their values of Lorentzian distance, $\mathscr S := (\mathcal S, d_{\ell})$, are there conditions (on set $\mathcal S$ and/or on the pairwise values $d_{\ell}$) under which it is possible to determine whether or not $\mathscr S$ is a "flat spacetime"?
If spacetime $\mathscr S$ satisfies such conditions, how exactly would be determined whether $\mathscr S$ is flat, or not?
And if $\mathscr S$ had thereby be found to be flat, then how can the corresponding values of "spacetime intervals, $s^2$" for all pairs of events in set $\mathcal S$ be expressed in terms of the given Lorentzian distance values $d_{\ell}$ ?
There exists a flat spacetime where every pair of events can be connected by both a spacelike path and by a timelike path. So in that case your "definition" would require that the distance between every pair of events is zero in that case. And that assignment of zero always works for any spacetime. So if you are claiming uniqueness you are wrong. We can just assert that every spacetime has a distance function that always assigns zero to any pair of events. You can't possibly get any information from that.
If instead I look at the spirit of your rules, many of the other properties seemed to be based on properties not obeyed by that spacetime. And it makes me think the whole theory was only designed to apply to some subset of spacetimes, probably time oriented ones. And that's a global property, so not accessible to experimental validation, so this is a math question.
And then it relates to the common concrete definition where the distance is just the supremum of zero and the lengths of all future directed (so only for time oriented spacetimes) causal paths from A to B.